Problem: Two circles have the same center $C.$ (Circles which have the same center are called concentric.) The larger circle has radius $10$ and the smaller circle has radius $6.$ Determine the area of the ring between these two circles. [asy]
import graph;
filldraw(circle((0,0),10), lightgray, black+linewidth(1));
filldraw(circle((0,0),6), white, black+linewidth(1));
dot((0,0));
label("$C$",(0,0),NE);
[/asy]
Answer: The area of a circle with radius $r$ is $\pi r^2.$

So the area of the larger circle is $\pi(10^2)=100\pi$ and the area of the smaller circle is $\pi(6^2)=36\pi.$

The area of the ring between the two circles is the difference of these two areas.

Therefore, the area of the ring is $100\pi - 36\pi = \boxed{64\pi}.$